Correlators in phase-ordering from Schrödinger-invariance

TL;DR

This work shows that ageing and scaling in phase-ordering kinetics after a quench to $T<T_c$ with a non-conserved order parameter can be derived from Schrödinger-invariance of four-point response functions. By adopting a non-equilibrium representation of the Schrödinger algebra and focusing on the covariance of the four-point correlator $\langle \phi\phi\tilde{\phi}\tilde{\phi}\rangle$, the authors obtain universal aging forms for two-time and single-time correlators, relate the autocorrelation exponent $\lambda$ to the passage exponent $\zeta_p$, and reproduce Porod's law and the bounds $\frac{d}{2} \leq \lambda \leq d$. They also establish finite-size and global-scaling results, including the extension $\lambda = d - 2\Theta$ of the Janssen–Schaub–Schmittmann relation with the slip exponent $\Theta = \frac{1}{2}(d-\lambda)$, and provide explicit forms for four-point functions in Appendices A and B. The approach highlights the universality of ageing phenomena by deriving key scaling relations from dynamical symmetry alone, without model-specific details, while acknowledging that the explicit scaling functions $F_C$ are not fixed by these symmetries. Overall, the work offers a symmetry-based route to predicting ageing behaviour in phase-ordering and lays groundwork for extensions to more complex non-equilibrium settings.

Abstract

Systems undergoing phase-ordering kinetics after a quench into the ordered phase with $0<T<T_c$ from a fully disordered initial state and with a non-conserved order-parameter have the dynamical exponent ${z}=2$. The long-time behaviour of their single-time and two-time correlators, determined by the noisy initial conditions, is derived from Schrödinger-invariance and we show that the generic ageing scaling forms of the correlators follow from the Schrödinger covariance of the four-point response functions. The autocorrelation exponent $λ$ is related to the passage exponent $ζ_p$ which describes the time-scale for the cross-over into the ageing regime. Both Porod's law and the bounds $d/2 \leq λ\leq d$ are reproduced in a simple way. The dynamical scaling in fully finite systems and of global correlators is found and the low-temperature generalisation $λ= d-2Θ$ of the Janssen-Schaub-Schmittmann scaling relation is derived.

Figures (1)

• Figure 1: Schematic illustration of physical ageing in single-time and two-time correlators. A typical single-time correlator $C(s;r)$ is in panel (a) for different times $s_1<s_2<s_3$. In panel (b), their collapse when replotted over against rescaled lengths $r/\ell(s)$ is shown where $\ell(s)\sim s^{1/\mathpzc{z}}$ is the dynamical length scale. In panel (c) a typical two-time auto-correlator $C(s+\tau,s)$ is displayed over against $\tau=t-s$, for different waiting times $s_1<s_2<s_3$. These data collapse when replotted in panel (d) over against $y=t/s$. The inset shows the asymptotic power-law form $f_C(y)\sim y^{-\lambda_C/z}$.